Polytope of Type {4,6}
Play with this polytope as a twisty puzzle
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6}*1296a
if this polytope has a name.
Group : SmallGroup(1296,2909)
Rank : 3
Schlafli Type : {4,6}
Number of vertices, edges, etc : 108, 324, 162
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Halving Operation
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {4,6}*432a, {4,6}*432b
6-fold quotients : {4,6}*216
9-fold quotients : {4,6}*144
18-fold quotients : {4,6}*72
27-fold quotients : {4,6}*48a
54-fold quotients : {2,6}*24
81-fold quotients : {4,2}*16
108-fold quotients : {2,3}*12
162-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 2.
81 facets:
81 of {4}*8
54 vertex figures:
54 of {6}*12
P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 2.
81 facets:
81 of {4}*8
57 vertex figures:
51 of {6}*12
6 of {3}*6
P/N, where N=<s0*s1*s0*s1> of order 2.
90 facets:
18 of {2}*4
72 of {4}*8
54 vertex figures:
54 of {6}*12
P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 3.
54 facets:
54 of {4}*8
36 vertex figures:
36 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 3.
54 facets:
54 of {4}*8
36 vertex figures:
36 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 3.
54 facets:
54 of {4}*8
36 vertex figures:
36 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1> of order 3.
54 facets:
54 of {4}*8
36 vertex figures:
36 of {6}*12
P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1> of order 3.
54 facets:
54 of {4}*8
42 vertex figures:
33 of {6}*12
9 of {2}*4
P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
45 facets:
9 of {2}*4
36 of {4}*8
30 vertex figures:
24 of {6}*12
6 of {3}*6
P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
45 facets:
9 of {2}*4
36 of {4}*8
27 vertex figures:
27 of {6}*12
P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2> of order 6.
36 facets:
18 of {2}*4
18 of {4}*8
18 vertex figures:
18 of {6}*12
P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s0*s1*s2> of order 6.
36 facets:
18 of {2}*4
18 of {4}*8
18 vertex figures:
18 of {6}*12
P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s2*s1*s0*s2*s1> of order 6.
27 facets:
27 of {4}*8
21 vertex figures:
15 of {6}*12
6 of {3}*6
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 9.
18 facets:
18 of {4}*8
12 vertex figures:
12 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 9.
18 facets:
18 of {4}*8
16 vertex figures:
10 of {6}*12
6 of {2}*4
P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s2*s1*s2> of order 12.
18 facets:
9 of {2}*4
9 of {4}*8
11 vertex figures:
4 of {3}*6
7 of {6}*12
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)(10,28)(11,30)(12,29)(13,31)(14,33)(15,32)(16,34)(17,36)(18,35)(19,55)(20,57)(21,56)(22,58)(23,60)(24,59)(25,61)(26,63)(27,62)(37,38)(40,41)(43,44)(46,66)(47,65)(48,64)(49,69)(50,68)(51,67)(52,72)(53,71)(54,70)(73,74)(76,77)(79,80);;
s1 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,64)(38,66)(39,65)(40,70)(41,72)(42,71)(43,67)(44,69)(45,68)(46,73)(47,75)(48,74)(49,79)(50,81)(51,80)(52,76)(53,78)(54,77);;
s2 := ( 1,42)( 2,40)( 3,41)( 4,39)( 5,37)( 6,38)( 7,45)( 8,43)( 9,44)(10,32)(11,33)(12,31)(13,29)(14,30)(15,28)(16,35)(17,36)(18,34)(19,49)(20,50)(21,51)(22,46)(23,47)(24,48)(25,52)(26,53)(27,54)(55,69)(56,67)(57,68)(58,66)(59,64)(60,65)(61,72)(62,70)(63,71)(73,76)(74,77)(75,78);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;; s1 := F.2;; s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(81)!( 2, 3)( 5, 6)( 8, 9)(10,28)(11,30)(12,29)(13,31)(14,33)(15,32)(16,34)(17,36)(18,35)(19,55)(20,57)(21,56)(22,58)(23,60)(24,59)(25,61)(26,63)(27,62)(37,38)(40,41)(43,44)(46,66)(47,65)(48,64)(49,69)(50,68)(51,67)(52,72)(53,71)(54,70)(73,74)(76,77)(79,80);
s1 := Sym(81)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,64)(38,66)(39,65)(40,70)(41,72)(42,71)(43,67)(44,69)(45,68)(46,73)(47,75)(48,74)(49,79)(50,81)(51,80)(52,76)(53,78)(54,77);
s2 := Sym(81)!( 1,42)( 2,40)( 3,41)( 4,39)( 5,37)( 6,38)( 7,45)( 8,43)( 9,44)(10,32)(11,33)(12,31)(13,29)(14,30)(15,28)(16,35)(17,36)(18,34)(19,49)(20,50)(21,51)(22,46)(23,47)(24,48)(25,52)(26,53)(27,54)(55,69)(56,67)(57,68)(58,66)(59,64)(60,65)(61,72)(62,70)(63,71)(73,76)(74,77)(75,78);
poly := sub<Sym(81)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;
References : None.
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